S5B1 - Graduate Seminar on Advanced Topics in PDE
(summer term 2015)

We will discuss in this seminar various topics in Analysis and PDE. The seminar may lead to research topics in the area that can lead to
a Master or PhD thesis. Students interested in participating should contact one of the organizers.

A first organizatorial meeting will be Friday April 10, 2015.

Talks (updated periodically):

April 17, 2015: Diogo Oliveira e Silva (University of Bonn)

Title: Restriction, Kakeya, Decoupling: an invitation.

Abstract: This is the first talk of a semester-long seminar geared towards an
understanding of Bourgain-Demeter's recent breakthrough, "The proof of the
l^2 decoupling conjecture". After a brief introduction to the restriction
and Kakeya problems, I will focus on their multilinear counterparts and try
to illustrate the various connections in play. I will state the decoupling
inequality and, time permitting, will list several applications to
analysis, incidence geometry and number theory.
This talk is intended as a non-technical survey; as such, many topics will
be discussed but few details will be provided.

A list of relevant literature (which could be used in the preparation of
future talks) follows:

Abstract: I will present the paper "A sharp bilinear restriction
estimate for paraboloids" by Tao. I will also discuss the trilinear
restriction estimate by Bennett, Carbery and Tao in the paper "On the
multilinear restriction and Kakeya conjectures".

Abstract: I will present part of the paper ``Bounds on
oscillatory integral operators based on multilinear estimates'' by
Bourgain and Guth. In this paper, the authors managed to apply the
multilinear restriction estimates to obtain certain improvement on the
linear restriction estimates.

May 15, 2015: Annegret Burtscher (University of Bonn)

Title: Geodesic incompleteness in general relativity.

Abstract: The general theory of relativity describes the effect of gravitation in
terms of the geometry of spacetimes via the Einstein equations. In the
1950s the initial value formulation and local existence of solutions to
the Einstein equations were established. As of yet the global structure of
solutions is much less understood, in general, singularities seem
unavoidable. The Penrose singularity theorems give some glimpse of this
singular nature by relating geodesic incompleteness to the existence of
trapped surfaces. In this talk we will see how such trapped surfaces can
form during evolution from regular initial data, illustrated for
spherically symmetric solutions of the Einstein-Euler equations.

June 5, 2015: Mariusz Mirek (University of Bonn)

Title: A local T(b) theorem for perfect multilinear Calder\'{o}n--Zygmund
operators

Abstract:
We will discuss a multilinear local T(b) theorem that differs from
previously considered multilinear local T(b) theorems in using
exclusively general testing functions b as opposed to a mix of general
testing functions and indicator functions. The main new feature is a set
of relations between the various testing functions b that to our
knowledge has not been observed in the literature and is necessitated by
our approach. For simplicity we restrict our attention to the perfect
dyadic model. This is a joint work with Christoph Thiele.

June 19, 2015: Anna Kosiorek

Title: A short proof of the multilinear Kakeya Inequality.

Abstract:
I will present a proof of a slightly weaker version of
multilinear Kakeya inequality, a geometric estimate about the overlap
pattern of cylindrical tubes in R^n pointing in different directions.
The talk will be based on a recent paper by Larry Guth.

June 26, 2015: Stefan Steinerberger (Yale University)

Title: Nonlinear phase unwrapping of a function.

Abstract:
One way of getting Fourier series is to take a holomorphic function,
create a root in the origin by translation, factor it out and repeat.
A nonlinear analogue would be to remove all roots inside the unit disk
and repeat - this gives rise to an unwinding series with many nice
properties. This series has been independently discovered by several
people and is easy to compute and very useful. We provide the first
proof of convergence in suitable spaces. This is joint work with Raphy
Coifman.

Abstract:
We will focus on the very recent paper "The proof of the
\ell^2 Decoupling Conjecture" by J. Bourgain and C. Demeter (Ann. of
Math. 2015). The proof relies on certain geometric/analytic techniques
developed in Bourgain-Guth's "Bounds on oscillatory integral operators
based on multilinear estimates" (Geom. Funct. Anal. 2011). In the first
half of this talk, Shaoming will recall some of these techniques. After the
break, Diogo will specialize to the two-dimensional situation and run the
induction-on-scales argument to finish the proof.

July 10, 2015: Blazej Wrobel (Universita di Milano)

Title: Approaching bilinear multipliers via a functional calculus

Abstract:
Bilinear multipliers for the Fourier transform may be defined in terms of
a joint functional calculus for partial derivatives. Using this
observation we propose a spectral generalization of the theory of bilinear
multipliers outside of the Fourier transform framework. We focus on
Coifman-Meyer type multiplier theorems and their relations with fractional
Leibniz rules. Examples admitted by our theory include bilinear
multipliers for: the discrete Laplacian, the general Dunkl Laplacian, and
the Jacobi operator. The talk is mostly based on work in progress.